DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Pell-equivalence for generalized solutions

Definition

Fix a generalized Pell equation $x^{2} - D y^{2} = N, N \neq 0,$ as in Generalized and negative Pell equations, and write $ε_{D} : = x_{1} + y_{1} \sqrt{D}$ for the fundamental Pell solution from The fundamental Pell solution.

Two integral solutions $α, β \in Z [\sqrt{D}], N_{D} (α) = N_{D} (β) = N,$ are Pell-equivalent if $β = α ε_{D}^{k}$ for some integer $k$ .

This stays inside the same generalized Pell equation because The Pell norm is multiplicative gives $N_{D} (α ε_{D}^{k}) = N_{D} (α) N_{D} (ε_{D})^{k} = N,$ and Integral Pell solutions form an abelian group makes the powers $ε_{D}^{k}$ a group.

Depends on

Used by

Generalized Pell solutions fall into finitely many Pell orbits Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Keith Conrad, Pell's Equation, II (standard reference, not scraped)
Peter Hackman, Elementary Number Theory (standard reference, not scraped)